Deterministic Finite Automaton

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DFA refers to deterministic finite automata. Deterministic refers to the uniqueness of the computation. The finite automata are deterministic FA, if the machine reads an input string one symbol at a time.

In DFA, there is only one path input from the current state to the next state. It does not accept the null move, i.e. it cannot change state without any input. It can contain multiple final states. It is used in Lexical Analysis in compilers.

Finite Automaton can be classified into two types −

• Deterministic Finite Automaton (DFA)
• Non-deterministic Finite Automaton (NDFA / NFA)

Deterministic Finite Automaton (DFA)

In DFA, for each input symbol, one can determine the state to which the machine will move. Hence, it is called Deterministic Automaton. As it has a finite number of states, the machine is called Deterministic Finite Machine or Deterministic Finite Automaton.

Formal Definition of a DFA

A Deterministic Finite automata (DFA) is a collection of defined as a 5-tuples and is as follows −

$$\mathrm{M \:=\:(Q,\: \Sigma,\: \delta, \:q0,\:F)}$$

Where,

• Q is a finite set of states.
• ∑ is a finite set of symbols called the alphabet.
• δ is the transition function where δ: Q × ∑ → Q
• q₀ is the initial state from where any input is processed (q₀ ∈ Q).
• F is a set of final state/states of Q (F ⊆ Q).

Graphical Representation of a DFA

A DFA is represented by digraphs called state diagram.

• The vertices represent the states.
• The arcs labeled with an input alphabet show the transitions.
• The initial state is denoted by an empty single incoming arc.
• The final state is indicated by double circles.

Example 1

Let a deterministic finite automaton be →

• Q = {a, b, c},
• ∑ = {0, 1},
• q₀ = {a},
• F = {c}, and

Transition Table

Transition function δ as shown by the following table −

Its graphical representation would be as follows −

Example 2

Minimize the following DFA

Solution

Make a transition Table.

$$\mathrm{\pi_0 \:=\: \{\{5\},\: \{1,\: 2,\: 3,\: 4\}\}}$$

$$\mathrm{\text{For input a, on }\: \{1,\:,2,\:3,\:4\} \: of \: \pi_0 }$$

For input b, on {1, 2, 3, 4} of π₀

$$\mathrm{\therefore \: \{1,\: 2,\: 3,\: 4\}\: \text{ will be split into } \: \{1, 3\} \: and \: \{2, 4\}}$$

$$\mathrm{\therefore \: \pi_1 \:=\: \{\{5\}, \:\{1, \:3\},\:\{2,\:4\}\}}$$

For input symbol a on {1, 3} of π₁

Similarly for input symbol a on {2, 4} of π₁

For input symbol b on {1, 3} of π₁

Similarly for input symbol b on {2, 4} of π₁

Subset in π₁ i.e., {1, 3} & {2, 4} will not be splitted.

$$\mathrm{\pi_{final} \:=\: \{\{5\},\: \{1,\: 3\},\: \{2,\: 4\}\}}$$

There will be 3 states of DFA.

$$\mathrm{\{5\},\: \{1,\: 3\} \:and \:\{2,\: 4\}}$$

Minimized DFA will be −

Example 3

Given a Deterministic Finite Automata (DFA), try to reduce the DFA by removing unreachable states and removing similar rows.

Solution

Step 1 − Remove the unreachable states from q0

From the initial states, we are not able to reach q2 and q4. So, remove these two states as shown below −

After removing unreachable states, the partial minimized DFA is as follows −

Step 2 − The transition table is given below −

Step 3 − Divide tables into 2 tables as shown below −

Table 1 starts from the non-final states.

Table 2 starts from the final states.

Step 4 − Remove similar rows.

Table 1 has no similar rows

Table 2 has similar rows. So, skip q5 and replace q5 by q3

Step 5 − Combine two tables as shown below −

Thus, the minimized DFA will be as follows −

Trap State in DFA

If a transition goes to a state from which it can never escape. Such a state is called a trap state. It is called the dead state.

In the above example, q2 is a trap or dead state because it can't reach the final state.

Application of Deterministic Finite Automata (DFA)

The different applications of deterministic finite automata are as follows −

• Protocol analysis text parsing.
• Video game character behavior.
• Security analysis.
• CPU control units.
• Natural language processing Speech recognition, etc.